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Viscoelastic fluids in thin domains: a mathematical
proof
Guy Bayada, Laurent Chupin, Bérénice Grec
To cite this version:
Viscoelastic fluids in thin domains: a mathematical proof
Guy Bayada1,2, Laurent Chupin2and B´er´enice Grec3,* Batiment L´eonard de Vinci - 21, avenue Jean Capelle
69 621 Villeurbanne cedex - France
Abstract
The present paper deals with non Newtonian viscoelastic flows of Oldroyd-B type in thin domains. Such geometries arise for example in the context of lubrication. More precisely, we justify rigorously the asymptotic model obtained heuristically by proving the mathematical convergence of the Navier-Stokes/Oldroyd-B sytem towards the asymptotic model.
Keywords: Viscoelastic fluids, Thin film, Oldroyd model, Lubrication flow, Asymptotic analysis.
1
Introduction
This paper concerns the study of a viscoelastic fluid flow in a thin gap, the motion of which is imposed due to non homogeneous boundary conditions.
When a Newtonian flow is contained between two close given surfaces in relative motion, it is well known that it is possible to replace the Stokes or Navier-Stokes equations governing the fluid’s motion by a simpler asymptotic model. The asymptotic pressure is proved to be independent of the normal direction to the close surfaces and obeys the Reynolds thin film equation whose coefficients include the velocities, the geometrical description of the surrounding surfaces and some rheological characteristics of the fluid. As a following step, the computation of this pressure allows an asymptotic velocity of the fluid to be easily computed. Such asymptotic procedure first proposed in a formal way by Reynolds [2] has been rigorously confirmed for Newtonian stationary flow [1], and then generalized in a lot of situations covering numerous applications for both compressible fluid [14], unsteady cases [3], multifluid flows [15].
It is well known however that in numerous applications, the fluid to be considered is a non Newtonian one. This is the case for numerous biological fluids, modern lubricants in engineering applications due to the additives they contain, polymers in injection or molding process. In all of these applications, there are situations in which the flow is anisotropic. It is usual to take
1
INSA-Lyon - Institut Camille Jordan - CNRS UMR 5208
2
INSA-Lyon - LAMCOS - CNRS UMR 5259
3
Ecole Centrale de Lyon - Institut Camille Jordan - CNRS UMR 5208
*
account of this geometrical effect in order to simplify the three-dimensional equations of the motion, trying to recover two dimensional Reynolds like equation with respect to the pressure only. Such procedures are more often heuristic ones. Nevertheless, some mathematical works appeared in the literature to justify them. They include thin film asymptotic studies of Bingham flow [9], quasi Newtonian flow (Carreau’s law, power law or Williamson’s law, in which various stress-velocity relations are chosen: [7], [6], [16]) and also micro polar ones [5]. It has been possible to obtain rigorously some thin film approximation for such fluids using a so called generalized Reynolds equation for the pressure.
However in the preceding examples, elasticity effects are neglected. Introduction of such viscoelastic behavior is characterized by the Deborah number which is related to the relaxation time. One of the most popular laws is the Oldroyd-B model whose constitutive equation is an interpolation between purely viscous and purely elastic models, thus introducing an additional parameter which describes the relative proportion of both behaviors. A formal procedure has been proposed in [4]. However, the asymptotic system so obtained lacks the usual characteristic of classical generalized Reynolds equation as it has not been possible to gain an equation in the asymptotic pressure only. Both velocity u∗ and pressure p∗ are coupled by a non linear system.
It is the goal of this paper to justify rigorously this asymptotic system. Section 2 is devoted to the precise statement of the 3-D problem. One difficulty has been to find an existence theorem for the general Oldroyd-B model, acting as a starting point for the mathematical procedure. Most of the existence theorems, however, deal with small data or small time assumptions. To control this kind of property with respect to the smallness of the gap appears somewhat difficult. So we are led to consider a more particular Oldroyd-B model, for which unconditional existence theorem has been proved [13]. Moreover, a specific attention is devoted to the boundary conditions to be introduced both on the velocity and on the stress. The goal is to use ”well prepared” boundary conditions so as to prevent boundary layer on the lateral side of the domain.
In Section 3, after suitable scaling procedure, asymptotic expansions of both pressure, viscosity and stress are introduced, taking into account the previous formal results from [4]. Section 4 is mainly concerned with the proof of some additional regularity properties for the formal asymptotic solution. Assuming some restrictions on the rheological parameters, it will be proved that it is possible to gain a Ck regularity for p∗ , k > 1, which in turn improves the regularity of u∗ and the
stress tensor σ∗. This result is obtained by introducing a differential Cauchy system satisfied by
the derivative of p∗. Finally, section 5, is devoted to the convergence towards zero of the second
term of the asymptotic expansions, which in turn proves the convergence of the solution of the real 3-D problem towards u∗, p∗, σ∗ (Theorems 5.4 and 5.6).
2
Introduction of the problem and known results
2.1 Formulation of the problem
domain, with n = 2 or n = 3 (x = x1 or x = (x1, x2)), as in Figure 1.
Figure 1: Domain ˆΩε
The following hypotheses on h are required:
∀x ∈ ω, 0 < h0 ≤ h(x) ≤ hεM, and hε∈ C1(¯ω).
Let ˆuε = (ˆuε1, ˆuε2, ˆuε3) be the velocity field in the three-dimensional case, or ˆuε = (ˆuε1, ˆuε2) in the two-dimensional case, ˆpε the pressure, and ˆσε the stress symmetric tensor in the domain ˆΩε.
Bold letters stand for vectorial or tensorial functions, the notation ˆf corresponds to a function f defined in the domain ˆΩε, and the superscript ε denotes the dependence on ε.
Formulation of the problem The following formulation of the problem holds in (0, ∞) × ˆΩε:
ρ ∂tuˆε+ ρ ˆuε· ∇ ˆuε− (1 − r)ν ∆ ˆuε+ ∇ˆpε = ∇ · ˆσε, ∇ · ˆuε = 0 , λ (∂tσˆε+ ˆuε· ∇ˆσε+ g( ˆσε, ∇ ˆuε)) + ˆσε = 2rνD( ˆuε) , (2.1)
where the nonlinear terms g( ˆσε, ∇ ˆuε), the vorticity tensor W ( ˆuε) and the deformation tensor D( ˆuε) are given by:
g( ˆσε, ∇ ˆuε) = −W ( ˆuε) · ˆσε+ ˆσε· W ( ˆuε), W ( ˆuε) = ∇ ˆu ε−t ∇ ˆuε 2 and D( ˆu ε) = ∇ ˆuε+ t ∇ ˆuε 2 .
Initial conditions This problem is considered with the following initial conditions: ˆ
uε|t=0= ˆuε0, σˆε|t=0= ˆσε0, (2.2)
for ˆuε0∈ L2( ˆΩε), ˆσ0ε∈ L2( ˆΩε). The bold notation L2( ˆΩε) denotes the set of vectorial or tensorial functions whose all components belong to L2( ˆΩε).
Boundary conditions Dirichlet boundary conditions are set on top and bottom of the domain, and the conditions on the lateral part of the boundary ˆΓε
L, defined by
ˆ
ΓεL= {(x, y) ∈ Rn, x ∈ ∂ω and 0 < y < εh(x)} ,
will be specified later (in section 4.2). Therefore, it is possible to write the boundary conditions in a shortened way:
ˆ
uε|∂ ˆΩε = ˆJε, (2.3)
where ˆJεis a given function such that ˆJε∈ H1/2(∂ ˆΩε) and satisfying ˆJε|
y=hε = 0, ˆJε|y=0= (s, 0).
This function will be fully determined in Subsection 4.2.
Since ˆσεsatisfies a transport equation in the domain ˆΩε, it remains to impose boundary conditions on ˆσε on the part of the boundary where ˆuε is an incoming velocity. Let us define ˆΓε
+ the part of ˆΓεL such that ˆJε|Γˆε + · n < 0, and ˆΓ ε −= ˆΓεL\ ˆΓε+. We set ˆ σε|Γˆε + = ˆθ ε, (2.4)
where ˆθε is a given function in H1/2(ˆΓε+) which will also be determined in Subsection 4.2. Moreover, since the pressure is defined up to a constant, the mean pressure is chosen to be zero:
R
ˆ Ωε
ˆ pε= 0.
Notations Let us introduce the following function space: V =nϕˆ∈ H01( ˆΩε), ∇ · ˆϕ= 0o,
and the following notations, that will be used in the following. For ˆf defined in ˆΩε: • | ˆf | denotes the L2-norm in ˆΩε,
• | ˆf |p denotes the Lp-norm in ˆΩε, for 2 < p ≤ +∞,
• the spaces Cm( ˆΩε) for m ≥ 1 are equipped with the norms k ˆf k
Cm = | ˆf |∞+
m
P
i=1| ˆ
f(i)|∞.
For ˆf defined in R+ × ˆΩε, k ˆf kLα(Lβ) denotes the norm of the space Lα(0, ∞, Lβ( ˆΩε)), with
2.2 Existence theorem in the domain ˆΩε
Theorem 2.1. For ε > 0 fixed, problem (2.1)-(2.3) admits a weak solution ˆ
uε∈ L2loc(0, ∞, H1( ˆΩε)) , pˆε∈ Lloc2 (0, ∞, L2( ˆΩε)) , σˆε∈ C(0, ∞, L2( ˆΩε)) .
Proof. This result is proved in [13].
Remark 2.2. Let us emphasize that for the following, it is essential to know the global (in time) existence of a solution for problem (2.1)-(2.3). Other existence theorems have been proved for this problem, for example in [12], [11], [10], but these theorems are either local in time (on a time interval [0, Tε]), or a small data assumption is needed. In this work, these theorems cannot be
used, since there is no control on the behavior of Tε (or equivalently of the data) when ε tends to zero, in particular Tε may tend to zero.
Consequently, this work is restricted to the specific case treated in [13], taking one parameter of the Oldroyd model to be zero. In all generality, the non-linear term reads g(σ, ∇u) = −W (u) · σ+ σ · W (u) − a (σ · D(u) + D(u) · σ), which is called objective derivative. Here the parameter a is taken to be zero. This case corresponds to the so-called Jaumann derivative.
Remark 2.3. The following computations are made in the two-dimensional case (i.e. ω = (0, L) is a one-dimensional domain) for the sake of simplicity. However, note that except for the regularity obtained for the limit problem in Section 4.3, all estimates are independent of the dimension, thus the corresponding computations should apply to the three-dimensional case. Regularizing the system In the proof of the preceding theorem, the existence of a solution is achieved by regularization. Therefore, this study only concerns solutions obtained as the limit of a regularized problem approximating (2.1), in which an additional term −η∆ˆσεη is added to the Oldroyd equation, with η > 0 a small parameter. Here a regularization of the form −η∆(ˆσεη− ˆG) is chosen, with ˆG a symmetric tensor in H2( ˆΩε) independent of η and ε which will be precised
later. After obtaining the needed energy estimates uniformly in η, we will let η tend to zero. This approach allows to multiply the Oldroyd equation by ˆσεη, since ˆσεη is regular enough. Of course, one can choose another regularization which leads to energy estimates which are uniform in the regularization parameter.
Furthermore, because of the regularizing term, boundary conditions on the whole boundary are needed. Let us write ˆσεη|∂ ˆΩε = ˆθεη, where ˆθεη is now a function of H1/2(∂ ˆΩε), which will be
3
Asymptotic expansions
3.1 Renormalization of the domain
Introducing a new variable z = y
ε, the system (2.1) can be rewritten in a fixed re-scaled domain: Ω = {(x, z) ∈ Rn, x ∈ ω and 0 < z < h(x)} .
For a function ˆf defined in Ωε, f is defined in Ω by f (x, z) = ˆf (x, εz). For a function f ∈ Lp(Ω),
|f|p still denotes the Lp-norm in Ω, and similar notations hold for the other norms. Moreover,
the regularizing term η∆σεη is introduced. Denoting σεη = σ
εη 11 σ εη 12 σεη12 σ22εη !
, and similar notations for the components of G, it holds in (0, ∞) × Ω :
ρ δtuεη1 − (1 − r)ν ∆εuεη1 + ∂xpεη− ∂xσ11εη− 1 ε∂zσ εη 12 = 0 , ρ δtuεη2 − (1 − r)ν ∆εuεη2 + 1 ε∂zp εη− ∂ xσ12εη− 1 ε∂zσ εη 22 = 0 , ∇ε· uεη = 0 , λ δtσ11εη− ˜N (uεη, σ εη 12) + σεη11− η∆ε(σ11εη− G11) − 2rν∂xuεη1 = 0 , λ δtσεη12+ 1 2N (u˜ εη, σεη 11− σ22εη) + σ12εη− η∆ε(σεη12− G12) − rν ∂xuεη2 + 1 ε∂zu εη 1 = 0 , λ δtσ22εη+ ˜N (uεη, σεη12) + σεη22− η∆ε(σ22εη− G22) − 2rν1 ε∂zu εη 2 = 0 , (3.1) where the convective derivative δt is given by δt = ∂t+ uεη · ∇ε. The derivation operators are
defined as follows: ∇ε = ∂x, 1 ε∂z and ∆ε= ∂x2+ 1
ε2∂z2. The non-linear terms ˜N are given by
˜ N (u, f ) = ∂xu2− 1 ε∂zu1 f . 3.2 Asymptotic expansions
It has been proposed in [4] that when η, ε tend zero, (uεη, pεη, σεη) tends formally to a triplet (u∗, p∗, σ∗) satisfying a system that will be given later in (4.1). This analysis leads to the
intro-duction of the following asymptotic expansions:
uεη1 = u∗1+ v1εη and uεη2 = εu∗2+ εvεη2 , (3.2) pεη= 1 ε2p ∗ + 1 ε2q εη, (3.3) σεη = 1 εσ ∗ +1 ετ εη, (3.4) with σ∗ = σ ∗ 11 σ∗12 σ∗ 12 σ∗22 ! , and τεη = τ εη 11 τ εη 12 τ12εη τ22εη ! . If denoting u∗ = (u∗ 1, u∗2), and vεη = (vεη1 , vεη2 ), (3.2) becomes uεη = u∗+ vεη.
of the stress tensor are motivated by some mathematical and physical remarks. Classically, the pressure has to be of order 1/ε2 if the horizontal velocity is of order 1 (see [2] for the rigorous explanation). On the other hand, the stress tensor has to be of order 1/ε and the Deborah number λ of order ε in order to balance the Newtonian and non-Newtonian contribution in Oldroyd equation (see [4]). Hence; let λ = ελ∗.
A wise choice of the function G in the regularizing term is G = σ∗. The regularity of G in
H2(Ω) is proved by Theorem 4.4 (where it is proved that ∂2
xσ∗ ∈ C0( ¯Ω), ∂x∂zσ∗ ∈ C0( ¯Ω) and
∂z2σ∗ ∈ C1( ¯Ω), thus ∆σ∗
∈ L2(Ω)). A formal substitution of (3.2), (3.3), (3.4) in (3.1) leads to the following system:
ρ dtvεη1 − (1 − r)ν ∆εv1εη+ 1 ε2∂xq εη−1 ε∂xτ εη 11 − 1 ε2∂zτ εη 12 = ˜L εη 1 + 1 εC1+ 1 ε2C ′ 1, ρ dtvεη2 − (1 − r)ν ∆εv2εη+ 1 ε4∂zq εη − 1 ε2∂xτ εη 12 − 1 ε3∂zτ εη 22 = 1 ε2L˜ εη 2 + 1 ε3C2+ 1 ε4C ′ 2, ∇ · vεη = ∇ · u∗, λ∗ (dtτ11εη− N(vεη, τ12εη)) + 1 ετ εη 11 − η∆ετ11εη − 2rν∂xvεη1 = ˜Lεη11+ 1 εL˜ ′εη 11, λ∗ dtτ12εη+ 1 2N (v εη, τεη 11 − τ22εη) +1 ετ εη 12 − η∆ετ12εη− rν ∂xvεη2 + 1 ε∂zv εη 1 = ˜Lεη12+ 1 εL˜ ′εη 12, λ∗ (d tτ22εη+ N (vεη, τ12εη)) + 1 ετ εη 22 − η∆ετ22εη − 2rν ε ∂zv εη 2 = ˜Lεη22+ 1 εL˜ ′εη 22, (3.5) with the following notations: dt= ∂t+ vεη·∇ is the so-called convective derivative, the non-linear
terms N (vεη, f ) = ε∂xv2εη− 1 ε∂zv εη 1
f for f ∈ L2(Ω) and the following linear (with respect to vεη) and constant terms
˜ Lεη1 = −ρ vεη· ∇u∗1− ρ u∗· ∇vεη1 | {z } Lεη1 −ρ ∂tu∗1− ρ u∗· ∇u∗1+ (1 − r)ν∂x2u∗1, C1 = ∂xσ11∗ , C1′ = (1 − r)ν∂z2u∗1− ∂xp∗+ ∂zσ12∗ ; ˜ Lεη2 = −ρ ε2vεη· ∇u∗2− ρ ε2u∗· ∇v2εη | {z } Lεη2 − ρ ε2∂tu∗2− ρ ε2u∗· ∇u∗2+ ε2(1 − r)ν∂2xu∗2+ (1 − r)ν∂z2u∗2+ ∂xσ12∗ , C2 = ∂zσ∗22, C2′ = ∂zp∗.
˜ L′11εη = −λ∗(∂zu1∗τ12εη + ∂zvεη1 σ ∗ 12) | {z } L′εη11 −λ∗∂zu∗1σ ∗ 12− σ ∗ 11; ˜ Lεη22=Lεη22− λ∗(∂tσ∗22+ u∗· ∇σ22∗ + ε∂xu∗2σ∗12) + 2rν∂zu∗2, with Lεη22= −λ∗(ε∂xu∗2τ12εη+ ε∂xv2σ∗12+ vεη· ∇σ ∗ 22+ u ∗ · ∇τ22εη) , ˜ L′22εη = λ∗(∂zu1∗τ12εη+ ∂zvεη1 σ ∗ 12) | {z } L′εη 22 +λ∗∂zu∗1σ ∗ 12− σ ∗ 22 ˜ Lεη12= −λ ∗ 2 (ε∂xu ∗ 2(τ11εη− τ εη 22) + ε∂xv εη 2 (σ ∗ 11− σ∗22) + 2vεη· ∇σ∗12+ 2u∗· ∇τ12εη) | {z } Lεη12 −λ ∗ 2 (2∂tσ ∗ 12+ 2u ∗ · ∇σ12∗ + ∂xu∗2(σ ∗ 11− σ ∗ 22)) + rνε∂xu∗2, ˜ L′εη12 = −λ ∗ 2 (∂zu ∗ 1(τ11εη− τ εη 22) + ∂zv1εη(σ ∗ 11− σ∗22)) | {z } L′12εη +λ ∗ 2 ∂zu ∗ 1(σ∗11− σ∗22) − σ∗12+ rν∂zu∗1;
Note that the first order derivatives of σ∗ occur in the terms ˜Lεη and Cεη. It will be shown in
Theorem 4.4 that σ∗ has sufficient regularity.
Let us observe also that equations (3.5) are similar to (3.1), except for the linear terms on the right. Thus the energy estimates will be obtained similarly for both systems, multiplying Navier-Stokes equation by the velocity and Oldroyd equation by the stress tensor, and integrating over Ω.
4
Limit equations
4.1 Limit system
In an heuristic way, the following system of equations satisfied by u∗, p∗, σ∗ is infered from (3.5):
This system is equipped with the following boundary condition (Dirichlet condition on the upper and lower part of the boundary, flux imposed on the lateral part of the boundary):
u∗= 0 , for z = h(x), u∗= (s, 0) , for z = 0, h(x)R 0 u∗dz · n = Φ0 on ΓL. (4.2)
The compatibility condition reads R
∂ω
Φ0 = 0. Moreover, since p∗ is defined up to a constant, the
mean pressure is taken to be zero: R
Ω
p∗= 0.
Remark 4.1. Each equation of the preceding system (4.1) is obtained by cancelling the constant part (i.e. the part independent of vεη, qεη, τεη) of respectively C′
1, C2′, ∇ · u∗, ˜L′εη11, ˜L ′εη 12, ˜L
′εη 22.
4.2 Determination of the boundary conditions
Remark 4.2. The lateral boundary conditions on u∗ do not depend on the ones on uεη, but only
on the flux. Therefore, different boundary conditions on uεη corresponding to the same flux lead to the same limit problem. This is a classical fact when passing from a two-dimensional problem to a one-dimensional problem (or similarly from a three-dimensional problem to a two-dimensional one), and has already been observed in [2] for example. Here, in order to avoid boundary layers, uεη = u∗ is imposed on the lateral part of the boundary.
Similarly, any value of σεη on the boundary leads to the same limit problem. Again, in order to
avoid boundary layers, well-prepared boundary conditions are also chosen for σεη.
The preceding remark allows to define precisely the function Jε introduced in (2.3). Since u∗|ΓL ∈ H
1/2(Γ
L), it is possible to construct Jε ∈ H1/2(∂Ω) satisfying Jε|z=h = 0, Jε|z=0 =
(s, 0) and Jε| ΓL = u
∗
|ΓL. Therefore, the boundary conditions on u
εη become uεη = 0 , for z = h(x), uεη = (s, 0) , for z = 0, uεη = u∗ on Γ L.
Thus uεη|∂Ω= u∗|∂Ω, and vεη will satisfy zero boundary conditions: vεη|∂Ω= 0.
Moreover, since σ∗
∈ H1(Ω) (see Theorem 4.4 for this regularity result), θε can be defined as
follows: θε= σ∗|Γ+ ∈ H 1/2(Γ +). (4.3) Therefore σεη|Γ+ = σ ∗ |Γ+,
On the other part Γ−of the boundary, σεη is chosen such that σεη· n|Γ−= σ ∗ · n|Γ−, for example σεη|Γ−= σ ∗ |Γ−.
4.3 Existence of a solution to the limit problem
System (4.1)-(4.2) has already been studied in [4].
Theorem 4.3. Assume that r < 8/9. Then system (4.1)-(4.2) has a unique solution satisfying u∗ ∈ L2(Ω), ∂zu∗ ∈ L2(Ω), p∗∈ H1(ω), σ∗ ∈ L2(Ω). (4.4)
Proof. This result has been proved in [4].
This existence result is not sufficient for this study. Therefore, the following stronger regularity result is proved on the limit problem (4.1)-(4.2).
Theorem 4.4. Assume r < 2/9. If h ∈ Hk(ω), for k ∈ N∗, then the unique solution (u∗, p∗, σ∗)
of the system (4.1)-(4.2) satisfies
p∗∈ Ck+1(¯ω), u1∗, ∂zu∗1, ∂z2u∗1∈ Ck+1( ¯Ω), σ∗, ∂zσ∗ ∈ Ck+1( ¯Ω),
∂xu∗1∈ Ck( ¯Ω), u∗2, ∂zu∗2, ∂z2u∗2∈ Ck( ¯Ω), ∂xσ∗ ∈ Ck( ¯Ω),
∂xu∗2∈ Ck−1( ¯Ω).
(4.5)
Proof. Let us observe that system (4.1) can be expressed as a system on u∗
1, p∗ only. Using (4.1),
σ∗
11, σ22∗ can be expressed as functions of σ12∗ and ∂zu∗1. Indeed, from the fourth and the last
equations of (4.1), it holds that
σ22∗ = −σ∗11= λ∗∂zu∗1σ12∗ . (4.6)
Moreover, the divergence-free equation can be rewritten in order to eliminate u∗
2. Integrating this
equation between z = 0 and z = h, and using the fact that u∗
2|z=0 = u∗2|z=h = u∗1|z=h = 0, it follows: ∂x h Z 0 u∗1dz = 0. (4.7)
Thus, the system in u∗
1, p∗ can be written in the following form:
− ν(1 − r)∂z2u∗1− ∂zσ12∗ + ∂xp∗ = 0, with σ∗12= νr∂zu∗1 1 + λ∗2|∂ zu∗1|2 , ∂zp∗= 0, ∂x h Z 0 u∗1dz = 0, (4.8)
equipped with the boundary conditions stated in (4.2) and the condition R
Ω
For the sake of readability, the superscripts∗ are omitted in the rest of this section.
Denote q = ∂xp. Let φ ∈ C∞(R) defined by φ(t) = ν(1 − r)t +
νrt
1 + λ2t2. The first equation of
(4.8) becomes q = ∂z(φ(∂zu1)).
A simple study of function φ allows to show the following properties: 0 < ν 1 −9r 8 < |φ′ |∞ < ν, and φ(t) −−−−→ t→±∞ ±∞. (4.9)
Therefore the function φ is invertible, and ψ = φ−1belongs to C∞(R). Moreover, ψ is an increasing
function as φ. Integrating q = ∂z(φ(∂zu1)) with respect to z between 0 and z, the first equation
of (4.8) becomes:
φ(∂zu1(x, z)) = q(x) z + κ(x),
where κ(x) is a integration constant. Therefore, it follows that ∂zu1(x, z) = ψ(q(x) z + κ(x)).
Since u1|z=0= s, the integration between 0 and z of the preceding equation yields:
u1(x, z) = s +
Z z
0
ψ(q(x)t + κ(x))dt. (4.10)
The boundary condition u1|h(x) = 0 implies also:
Z h(x)
0
ψ(q(x)t + κ(x)) + s = 0. (4.11)
For (h, q, s, κ) ∈ R4, let us introduce F (h, q, s, κ) = Z h
0
ψ(qt + κ) + s.
Lemma 4.5. For any (h, q, s) ∈ R3 there exists an unique κ ∈ R such that F (h, q, s, κ) = 0. Proof. • If such an κ exists, it is unique from the implicit function theorem, since for all
(h, q, s, κ) ∈ R4 ∂F ∂κ(h, q, s, κ) = Z h 0 ψ′(qt + κ)dt > 0.
• The following limits are computed, using the fact that limt→±∞ψ(t) = ±∞:
lim
κ→+∞F (h, q, s, κ) = +∞ and κ→−∞lim F (h, q, s, κ) = −∞.
Therefore, the following expression holds for (h, q, s) ∈ R3:
F (h, q, s, K(h, q, s)) = 0. (4.12)
It is now possible to obtain an information on the sign of ∂qK. Indeed, deriving the expression
(4.12) with respect to q, it follows
∂qF + ∂κF ∂qK = 0. For h > 0, since ∂qF = Z h 0 tψ′(qt + κ)dt > 0 and ∂aF = Z h 0 ψ′(qt + κ)dt > 0, ∂qK is strictly negative.
Now, using equation (4.7) and the expression (4.10) for u, it follows: Z h(x) 0 Z z 0 ∂x ψ(q(x)t + K(h(x), q(x), s))dt dz = 0. or if changing the direction of integration
Z h(x)
0 (h(x) − t)∂ x
ψ(q(x)t + K(h(x), q(x), s))dt = 0. This can be rewritten as
q′(x) Z h(x) 0 (h(x) − t) (t + ∂qK(h(x), q(x), s) ψ′ q(x)t + K(h(x), q(x), s)dt = − Z h(x) 0 (h(x) − t) h′(x)∂ hK(h(x), q(x), s) ψ′ q(x)t + K(h(x), q(x), s)dt,
which can be seen as an ordinary differential equation in q. Let U (x, q) = Z h(x) 0 h(x) − t t + ∂qK(h(x), q, s) ψ′ qt + K(h(x), q, s)dt, V (x, q) = Z h(x) 0 h(x) − t h′(x)∂hK(h(x), q, s) ψ′ qt + K(h(x), q, s)dt.
The differential equation becomes U (x, q(x)) q′
(x) = −V (x, q(x)) for x ∈ ω. Note that this equation is in some sense a generalized Reynolds equation for the pressure.
Lemma 4.6. Let r < 2/9. Then U (x, q) < 0 for any (x, q) ∈ ω × R.
Proof. Let (x, q) ∈ ω × R. Equation (4.11) and the definition (4.12) of K imply: Z h(x)
which becomes, after derivation with respect to q Z h(x) 0 t + ∂qK(h(x), q, s) ψ′ qt + K(h(x), q, s)dt = 0. (4.13)
With the notation K′(x, q) = ∂
qK(h(x), q, s), (4.13) implies K′(x, q) = − Z h(x) 0 t ψ′ qt + K(h(x), q, s)dt Z h(x) 0 ψ′ qt + K(h(x), q, s)dt .
Now, using this expression, U (x, q) can be simplified: U (x, q) = Z h(x) 0 −t t + ∂qK(h(x), q, s) ψ′ qt + K(h(x), q, s)dt. (4.14)
Recalling the estimate of |φ|∞ in (4.9), it follows that for any t ∈ R:
1 ν < ψ ′ (t) = 1 φ′(ψ(t) < 1 ν(1 − 9r/8) Let m = 1 ν, M = 1 ν(1 − 9r/8). Then −bh(x)2m ≤ K′(x, q) ≤ −ah(x)2M .
Now, (4.14) implies that: h(x)3 m 3 − M 4 = Z h(x) 0 tm t − M h(x) 2m ≤ −U(x, q) ≤ Z h(x) 0 tM t − mh(x) 2M = h(x)3 M 3 − m 4 .
In order to prove that U remains strictly negative, it suffices to prove that 0 < m 3 − M 4 , i.e. that m M > 3
4, which is satisfied under the condition r < 2 9.
It is possible to apply Picard-Lindel¨of theorem (or Cauchy-Lipschitz theorem) to the ordinary differential equation −U(x, q(x)) q′(x) = V (x, q(x)), as U remains strictly negative by Lemma
4.6. Since ψ and K are C∞-functions, the regularity of q′ is determined by the regularity of q and
h. By hypothesis, h belongs to Hk(ω), with k ∈ N, hence h ∈ L2(ω). Moreover, Theorem 4.3 implies that q ∈ L2(ω). Thus q′
∈ L2(ω), which means q ∈ H1(ω).
Iterating this process as long as h is regular, h ∈ Hk(ω) and q ∈ Hk(ω) implies that q′
∈ Hk(ω),
thus ∂xp = q ∈ Hk+1(ω), and p ∈ Hk+2(ω). By the classical Sobolev embedding, p belongs to
Last, recalling the expression (4.10), it follows that u1 ∈ Ck+1(¯ω), and, taking the first and second
derivatives of (4.10) with respect to z, that ∂zu1, ∂z2u1 also belong to Ck+1(¯ω).
As observed in the introduction of the proof, σ and u2 are given as functions of p, u1, and the
needed regularity follows.
Remark 4.7. Since in practical applications, h is very regular (h ∈ C∞(¯ω)), the preceding theorem gives as much regularity as wanted. In particular, the following result will be useful subsequently.
Corollary 4.8. Assume r < 2/9. If h ∈ H1(ω), then the unique solution (u∗, p∗, σ∗) of the
system (4.1)-(4.2) satisfies p∗ ∈ C2(¯ω), u∗1, ∂zu∗1, ∂2zu ∗ 1 ∈ C2( ¯Ω), σ ∗ , ∂zσ∗ ∈ C2( ¯Ω), ∂xu∗1∈ C1( ¯Ω), u ∗ 2, ∂zu∗2, ∂z2u ∗ 2 ∈ C1( ¯Ω), ∂xσ∗∈ C1( ¯Ω), ∂xu∗2∈ C0( ¯Ω). (4.15)
Proof. It suffices to take k = 1 in the preceding theorem 4.4.
5
Convergence of the remainders
5.1 Equations on the remainders
From now on, the superscript εη are dropped although the functions still depend on ε and η. Using the equations (4.1), system (3.5) becomes
ρ dtv1− (1 − r)ν ∆εv1+ 1 ε2∂xq − 1 ε∂xτ11− 1 ε2∂zτ12= L1+ 1 εC1, (5.1a) ρ dtv2− (1 − r)ν ∆εv2+ 1 ε4∂xq − 1 ε2∂xτ12− 1 ε3∂zτ22= 1 ε2L2+ 1 ε3C2, (5.1b) ∇ · v = 0, (5.1c) λ∗dtτ11− λ∗N (v, τ12) + 1 ετ11− η∆ετ11− 2rν∂xv1 = L11+ 1 εL ′ 11+ η∆εσ11∗ , (5.1d) λ∗dtτ12+ λ∗ 2 N (v, τ11− τ22) + 1 ετ12− η∆ετ12− rν ∂xv2+ 1 ε∂zv1 = L12+ 1 εL ′ 12+ η∆εσ12∗ ,(5.1e) λ∗dtτ22+ λ∗N (v, τ12) + 1 ετ22− η∆ετ22− 2rν ε ∂zv2 = L22+ 1 εL ′ 22+ η∆εσ22∗ , (5.1f)
with the new quantities
L′12= L′12,
L22= L22− λ∗(u∗· ∇σ∗22+ ε∂xu∗2σ ∗
12) + 2rν∂zu∗2,
L′22= L′22.
and with the initial and boundary conditions
v|t=0= u0− u∗, τ|t=0= σ0− σ∗, v|∂Ω= 0, τ|Γ+ = 0. (5.2)
Let us observe that both initial conditions v|t=0and τ |t=0belong to L2(Ω). v, q and τ are defined
by (3.2), (3.3), (3.4). From the existence theorem 2.1 for (u, p, σ) and theorem 4.3 for (u∗, p∗, σ∗),
it follows that system (5.1) admits a solution (v, q, τ ) ∈ L2(0, ∞, H1(Ω)) × L2(0, ∞, L2(Ω)) × C(0, ∞, L2(Ω)) for r < 8/9.
5.2 Convergence of v and τ
Before starting the a priori estimates, let us explain how the non-linear terms in (5.1) are handled. The non-linear terms v · ∇v of Navier-Stokes equation and v · ∇τ of Oldroyd equation are treated with the following Lemma 5.1. On the other hand, the non-linear terms N (v, τ ) =
ε∂xv2− 1ε∂zv1τ in (5.1d)-(5.1f) are zero when multiplied by τ .
Lemma 5.1. Let n be the exterior normal of the domain Ω. Let φ ∈ H1(Ω) be a vector field satisfying ∇ · φ = 0 and φ · n|∂Ω= 0. Let w ∈ H1(Ω). Then
Z
Ω
φ· ∇w w = 0.
Proof. By integration by parts: Z Ω φ· ∇w w = − Z Ω ∇ · φ | {z } =0 ·w2− Z Ω φ· ∇w w + Z ∂Ω φ· n | {z } =0 w2 = 0.
The classical approach consists in obtaining a priori estimates for v.
Proposition 5.2. Let (v, q, τ ) be a solution of (5.1). Then v = (v1, v2) satisfy the following
inequality for ε small enough: rνρd dt |v1| 2+ |εv 2|2 + 3 2r(1 − r)ν 2 |∇ εv1|2+ |ε∇εv2|2 ≤ −D1− D2+ C, (5.3) where D1 = 2rν ε Z Ω τ11∂xv1+ 2rν ε2 Z Ω τ12∂zv1, D2 = 2rν Z Ω τ12∂xv2+ 2rν ε Z Ω τ22∂zv2 and C is a constant independent of ε.
Step 1. Let us multiply (5.1a) by v1 and integrate over Ω. Observe that v1 is regular enough
to do so. Since v|∂Ω= 0, the boundary terms in the integration by parts are all zero. For example
−R
Ω
∆εv1v1 =R Ω |∇
εv1|2. Moreover, the convection terms R Ω
v· ∇v1v1 contained in R Ω
dtv1v1 are
equal to zero by Lemma 5.1, since ∇ · v = 0 and v|∂Ω= 0. It follows:
ρ 2 d dt|v1| 2+ (1 − r)ν|∇ εv1|2− 1 ε2 Z Ω q ∂xv1= − 1 ε Z Ω τ11∂xv1− 1 ε2 Z Ω τ12∂zv1 | {z } −D1/2rν + Z Ω L1v1+ 1 ε Z Ω C1v1. (5.4) It remains to estimate the termsR
Ω
L1v1 and R Ω
C1v1.
Main idea Estimates of the form: R
Ω
L1v1+1ε
R
Ω
C1v1 ≤ C +κ1|∇εv1|2+κ2|∂zv2|2will be proved,
where C is a constant independent of ε and where the constants κ1, κ2 satisfy κ1, κ2 < (1 − r)ν/4.
These constants will be precised later in the proof.
In the following, C, ci and Mi will denote some constants independent of ε and η, which might
depend on |Ω|, on the physical parameters of the problem and on u∗, σ∗ in sufficiently regular
norms.
• Let us estimate first the linear (with respect to v) term L1of L1. To this end, Poincar inequality
is useful: for f ∈ L2(Ω), with f |
z=h= 0, |f| ≤ CP|∂zf |. The constant CP only depends on Ω.
⋆ ρ Z Ω v1∂xu∗1v1 ≤ ρ|∂xu∗1|∞|v1|2 ≤ ρ ε2CP2|∂xu∗1|∞ 1 ε∂zv1 2 =: M1ε2 1 ε∂zv1 2 .
Note that by Theorem 4.4, ∂xu∗1 ∈ L∞(Ω). In the following, all the regularity results used in
the estimates also follow from Theorem 4.4.
⋆ For the next term, Poincar inequality is combined with Young inequality: ρ Z Ω v2∂zu∗1v1≤ ρ|∂zu∗1|∞|v2| |v1| ≤ ρ CP2|∂zu∗1|∞|∂zv2| |∂zv1| ≤ ρ CP2|∂zu∗1|∞ | {z } =:M2 ε 2|∂zv2| 2+ε 2 1 ε∂zv1 2! . ⋆ In a similar way: ρ Z Ω u∗· ∇v1v1 ≤ ρ CP|u∗1|∞ | {z } =:M3 ε 2|∂xv1| 2+ ε 2 1 ε∂zv1 2! + ε2ρ CP|u∗2|∞ | {z } =:M4 1 ε∂zv1 2 .
Observe here that it was not possible to apply Lemma 5.1, since u∗
· n|∂Ω6= 0.
⋆ The first term is treated using again Poincar and Young inequalities: ρ Z Ω u∗·∇u∗1v1 ≤ ρ CP|u∗|∞|∇u∗1| |∂zv1| ≤ 1 2(ρ CP|u ∗ |∞|∇u∗1|)2+ ε2 2 1 ε∂zv1 2 ≤ C+ε 2 2 1 ε∂zv1 2 . ⋆ Similarly, (1 − r)ν Z Ω ∂x2u∗ 1v1 ≤ C + ε2 2 1 ε∂zv1 2 .
⋆ The last term is estimated as follows, using Young inequality: 1 ε Z Ω ∂xσ∗11v1≤ 1 4c|∂xσ ∗ 11|2+ c 1 ε∂zv1 2 ≤ C + c 1 ε∂zv1 2 ,
where c is a positive constant independent of ε that can be chosen arbitrarily. Now, let us choose ε and c small enough such that all constants satisfy:
M1ε2,M2ε 2 , M3ε 2 , M4ε 2,ε2 2 , c ≤ (1 − r)ν 36 . (5.5)
Step 2. Let us multiply (5.1b) by ε2v2 and integrate over Ω. Again, the boundary terms in
the integrations by parts vanish, since v2|∂Ω= 0, and the convection terms are equal to zero since
∇ · v = 0 and v|∂Ω= 0 (by Lemma 5.1). It follows:
ρ ε2 2 d dt|v2| 2+ (1 −r)ν|ε∇ εv2|2− 1 ε2 Z Ω q ∂zv2 = − Z Ω τ12∂xv2− 1 ε Z Ω τ22∂zv2 | {z } −D2/2rν + Z Ω L2v2+ 1 ε Z Ω C2v2. (5.6) Each term of R Ω L2v2 and R Ω
C2v2 is estimated with the help of Poincar and Young inequalities as
in the preceding step. ⋆ ε2ρ Z Ω v· ∇u∗2v2 ≤ ε2ρCP2|∂xu∗2|∞ | {z } =:M5 ε 2 1 ε∂zv1 2 +ε 2|∂zv2| 2 ! + ε2ρCP2|∂zu∗2|∞ | {z } =:M6 |∂zv2|2. ⋆ ε2ρ Z Ω u∗· ∇v2v2≤ ε ρCP|u∗1|∞ | {z } =:M7 |ε∂xv2|2+ |∂zv2|2+ ε2ρCP|u∗2|∞ | {z } =:M8 |∂zv2|2. ⋆ ε2ρ Z Ω u∗· ∇u∗2v2 ≤ 1 2ε 2ρ|u∗ |2∞|∇u∗2|2+ ε2 1 2C 2 P | {z } =:M9 |∂zv2|2 ≤ C + ε2M9|∂zv2|2.
⋆ By integration by parts (all boundary terms are equal to zero since v2|∂Ω = 0) and Young
⋆ (1 − r)ν Z Ω ∂2zu∗2r2 ≤ 1 4c1(1 − r) 2ν2C2 P|∂z2u ∗ 2|2+ c1|∂zv2|2≤ C + c1|∂zv2|2, where c1 is a arbitrary positive constant. ⋆ Z Ω ∂xσ12∗ v2≤ CP2 4c1|∂x σ∗12|2+ c1|∂zv2|2 ≤ C + c1|∂zv2|2.
⋆ The C2 term is treated with integration by parts (again, no boundary terms since v2|∂Ω =
v1|∂Ω= 0) and the divergence equation. The term is then treated as the preceding one:
1 ε Z Ω ∂zσ∗22v2 = − 1 ε Z Ω σ∗22∂zv2= 1 ε Z Ω σ22∗ ∂xv1 = − 1 ε Z Ω ∂xσ22∗ v1 ≤ CP|∂xσ22∗ | 1 ε∂zv1 ≤ CP2 4c2|∂x σ∗22|2+ c2 1 ε∂zv1 2 ≤ C + c2 1 ε∂zv1 2 .
Now, let us choose ε, c1 and c2 small enough such that
M5ε3 2 , M6ε 2, M 7ε, M8ε, M9ε, M10ε 2 , c1, c2 ≤ (1 − r)ν 36 . (5.7)
Step 3. After summing (5.4) and (5.6), and multiplying by 2rν, it holds for ε small enough (satisfying (5.5) and (5.7)): rνρd dt |v1| 2+ |εv 2|2 +3 2r(1−r)ν 2 |∇ εv1|2+ |ε∇εv2|2 −2rνε2 Z Ω q (∂xv1+ ∂zv2) ≤ −D1−D2+C,
where C is a constant independent of ε. From the divergence equation ∇ · v = ∂xv1+ ∂zv2 = 0
it follows that the pressure term R
Ω
q (∂xv1+ ∂zv2) = 0, and equation (5.3) is obtained.
Proposition 5.3. Let us suppose that
λ∗|∂zu∗1|∞≤ 1/12, λ∗|σ∗12|∞≤ χ, λ∗(|σ11∗ |∞+ |σ∗22|∞) ≤ χ, 2λ∗|∂zσ∗12|∞ ≤ χ, λ∗|∂zσ11∗ |∞≤ χ,
where χ = ν 6
p
r(1 − r). Then for ε small enough, τ11, τ12, τ22 solution of (5.1) satisfy the
following inequality: λ∗ 2ε d dt |τ11| 2 + 2|τ12|2+ |τ22|2 +1 2 1 ετ11 2 + 2 1 ετ12 2 + 1 ετ22 2! +η ε |∇ετ11| 2+ 2|∇ ετ12|2+ |∇ετ22|2≤ D1+ D2+ r(1 − r)ν2 |∇εv1|2+ |ε∇εv2|2+ C, (5.8) where C is a constant independent of ε.
Proof. As in the preceding proposition, classical a priori estimates on τ11, τ12 and τ22 are
Step 1. Let us multiply (5.1d) by τ11
ε and integrate over Ω. Again, the convection terms R
Ω
v · ∇τ11τ11 contained in
R
Ω
dtτ11τ11 are equal to zero by Lemma 5.1, since ∇ · v = 0 and
v|∂Ω = 0 (see (5.2)). Moreover, there is no boundary term in the integration by parts since the
boundary conditions on σ have be chosen such that τ · n|∂Ω= 0 (see also (5.2)). It follows:
λ∗ 2ε d dt|τ11| 2 −λ ∗ ε Z Ω N (v, τ12) τ11+ 1 ετ11 2 +η ε|∇ετ11| 2 = 2rν ε Z Ω ∂xv1τ11+ 1 ε Z Ω L11τ11+ 1 ε2 Z Ω L′11τ11. (5.9) • The terms of Z Ω
L11τ11 are estimated as follows:
⋆ λ∗ Z Ω ∂xu∗2τ12τ11≤ λ∗|∂xu∗2|∞ | {z } =:M11 ε2 2 1 ετ11 2 +ε 2 2 1 ετ12 2! . ⋆ In a same way: λ ∗ ε Z Ω v1∂xσ11∗ τ11≤ λ∗|∂xσ∗11|∞CP | {z } =:M12 ε 2 1 ε∂zv1 2 +ε 2 1 ετ11 2! . Let us choose ε small enough such that:
M11ε2 2 ≤ 1 24 and M12ε 2 ≤ Min r(1 − r)ν 6 , 1 24 . ⋆ λ∗ Z Ω ∂xv2σ∗12τ11≤ λ∗|σ12∗ |∞|ε∂xv2| 1 ετ11 ≤ λ∗|σ∗12|∞ 1 4c3|ε∂x v2|2+ c3 1 ετ11 2! .
Here, it is not possible to choose c3 such that both coefficients are less than r(1 − r)ν/6 and
1/24. Therefore,a condition on λ∗
|σ∗
12|∞ is imposed such that:
λ∗ |σ∗ 12|∞ 4c3 ≤ r(1 − r)ν 6 and λ ∗ |σ∗12|∞c3 ≤ 1 24. Choosing c3 satisfying λ∗|σ∗12|∞c3 = 1/24, the condition on λ∗|σ12∗ |∞ becomes:
λ∗|σ12∗ |∞≤
ν 6
p
r(1 − r) =: χ. ⋆ Similarly the following term can be estimated:
λ∗ ε Z Ω v2∂zσ∗11τ11≤ λ∗|∂zσ∗11|∞|∂zv2| 1 ετ11 ≤ λ∗|∂zσ∗11|∞ 1 4c3|∂z v2|2+ c3 1 ετ11 2! .
The same reasoning as before allows to control both terms providing that λ∗
|∂zσ∗11|∞≤ χ.
⋆ In order to treat the term −λ∗
Z
Ω
u∗· n|∂Ω6= 0. However, integration by parts implies that −λ∗ Z Ω u∗· ∇τ11τ11= − λ∗ 2 Z ∂Ω u∗· n τ112.
On ω, since u∗ = (s, 0) (see (4.2)), it holds u∗
· n = 0. Thus it remains to consider the boundary integral on ΓL. This boundary integral is split into two integrals on Γ+ and Γ−. On
Γ−, it holds u∗· n > 0, thus −λ
∗
2
R
Γ−
u∗· n τ112 ≤ 0, and this term is trivially bounded by zero. On Γ+, the boundary conditions are chosen in subsection 4.2 such that τ |Γ+ = 0, therefore
−λ2∗
R
Γ+
u∗· n τ112 = 0.
• All other terms of Z
Ω
L11τ11 are easier to manage, since they are linear in τ11, and they are
treated with Young and Poincar inequalities in a same way as the ones in v1, v2.
• For the terms of Z Ω L′11τ11, we proceed as before: λ∗ ε2 Z Ω ∂zu∗1τ12τ11≤ λ∗|∂zu∗1|∞ 1 ετ12 1 ετ11 ≤ λ∗|∂zu∗1|∞ 1 2 1 ετ12 2 +1 2 1 ετ11 2! . Choosing λ∗
|∂zu∗1|∞≤ 1/12, both terms are bounded by 1/24.
λ∗ ε2 Z Ω ∂zv1σ12∗ τ11≤ λ∗|σ12∗ |∞ 1 ε∂zv1 1 ετ11 ≤ λ∗|σ∗12|∞ 1 4c3 1 ε∂zv1 2 + c3 1 ετ11 2! . Imposing λ∗ |σ∗
12|∞≤ χ is enough to ensure that the coefficients are less than r(1 − r)ν/6 and
1/24.
Step 2. Now, multiplying equation (5.1e) by 2τ12
ε and integrating over Ω, with the same reasoning as in the preceding step it follows:
λ∗ ε d dt|τ12| 2+λ∗ ε Z Ω N (v, τ11− τ22) τ12+ 2 1 ετ12 2 +2η ε |∇ετ12| 2 = 2rν ε Z Ω ∂xv2+ 1 ε∂zv1 τ12+ 2 ε Z Ω L12τ12+ 2 ε2 Z Ω L′12τ12 (5.10)
The terms in L12 and L′12 are of the same type as the ones in L11 and L′11, and are treated very
similarly to them, applying Young inequality, and assuming smallness assumptions on ε. Thus, let us only write the terms needing additional assumptions.
⋆ λ∗ Z Ω ∂xv2(σ∗11− σ∗22) τ12≤ λ∗(|σ11∗ |∞+ |σ∗22|∞) |ε∂xv2| 1 ετ12
, and it is enough to assume that λ∗
(|σ∗
⋆ 2λ ∗ ε Z Ω v2∂zσ∗12τ12≤ 2λ∗|∂zσ∗12|∞|∂zv2| 1 ετ12
, and we assume that 2λ∗|∂zσ12∗ |∞≤ χ.
⋆ λ ∗ ε2 Z Ω ∂zu∗(τ11− τ22) τ12 ≤ λ∗|∂zu∗1|∞ 1 ετ11 + 1 ετ22 1 ετ12
, it has already been assumed that λ∗ |∂zu∗1|∞≤ 1/12. ⋆ λ ∗ ε2 Z Ω ∂zv1(σ∗11− σ∗22) τ12 ≤ λ∗(|σ11∗ |∞ + |σ∗22|∞) 1 ε∂zv1 1 ετ12
, it has already been assumed that λ∗
(|σ∗
11|∞+ |σ∗22|∞) ≤ χ.
Step 3. Multiplying (5.1f) by τ22
ε , and estimating the terms just as the ones in τ11, it follows
λ∗ 2ε d dt|τ22| 2+ λ∗ ε Z Ω N (v, τ12) τ22+ 1 ετ22 2 +η ε|∇ετ22| 2 = 2rν ε2 Z Ω ∂zv2c + 1 ε Z Ω L22τ22+ 1 ε2 Z Ω L′22τ22. (5.11) Assuming that λ|σ∗
12|∞≤ χ, λ∗|∂zσ11∗ |∞≤ χ and λ∗|∂zu∗1|∞≤ 1/12, all the terms
1 ε R Ω L22τ22 and 1 ε2 R Ω
L′22τ22 are bounded and estimated as in Step 1.
Step 4. Summing (5.9), (5.10) and (5.11), and noticing that − Z Ω N (v, τ12) τ11+ Z Ω N (v, τ11− τ22) τ12+ Z Ω N (v, τ12) τ22 = Z Ω ε∂xv2− 1 ε∂zv1 (−τ12τ11+ (τ11− τ22) τ12+ τ12τ22) = 0,
it follows that for ε small enough λ∗ 2ε d dt |τ11| 2 + 2|τ12|2+ |τ22|2+1 2 1 ετ11 2 + 2 1 ετ12 2 + 1 ετ22 2! +η ε |∇ετ11| 2 + 2|∇ετ12|2+ |∇ετ22|2 ≤ D1+ D2+ r(1 − r)ν2 |∇εv1|2+ |ε∇εv2|2+ C,
where we recognized the terms D1+ D2, and where C is a constant independent of ε.
From now on, let us come back to the notation with the superscriptsεη, denoting the dependence
on ε and η.
small-ness assumptions
λ∗|∂zu∗1|∞≤ 1/12, λ∗|σ∗12|∞≤ χ, λ∗(|σ11∗ |∞+ |σ∗22|∞) ≤ χ, 2λ∗|∂zσ∗12|∞ ≤ χ, λ∗|∂zσ11∗ |∞≤ χ,
(5.12) where χ = ν6pr(1 − r). Then the following convergences hold up to subsequences when η and then ε tend to zero:
uεη1 → u∗1, ∂zuεη1 → ∂zu∗1, ∂xuεη1 ⇀ ∂xu∗1 in L2(0, T, L2(Ω)), (5.13)
uεη2 → 0, ∂zuεη2 → 0, ∂xuεη2 ⇀ 0 in L2(0, T, L2(Ω)), (5.14)
εσεη → σ∗ in L2(0, T, L2(Ω)), (5.15)
uεη1 ⇀∗u∗1, uεη2 ⇀∗ 0, εσεη → σ∗ in L∞(0, T, L2(Ω)). (5.16)
Proof. Summing (5.3), (5.8) implies that for ε small enough (i.e. if assumption (5.12) is satisfied):
rνρd dt |v εη 1 |2+ |εv2εη|2 +λ ∗ 2ε d dt |τ εη 11|2+ 2|τ12εη|2+ |τ22εη|2 +η ε |∇ετ εη 11|2+ 2|∇ετ12εη|2+ |∇ετ22εη|2 +r(1 − r)ν 2 2 |∂xv εη 1 |2+ 1 ε∂zv εη 1 2 + |ε∂xv2εη|2+ |∂zv2εη|2 ! +1 2 1 ετ εη 11 2 + 1 ετ εη 12 2 +1 2 1 ετ εη 22 2 ≤ C. (5.17) From this inequality, it follows that vεη converges to vεin H1(Ω) and τεη converges τεin L2(Ω), as η tends to zero. vε and τε are the solutions solutions of (5.1) without the terms η∆τεη.
Indeed, recalling the weak formulation of the system (5.1), it suffices to notice that Hlder’s inequality allows to treat the term η∆τεη:
η Z Ω ∇ετεη· ∇εφ≤ η1/2 η|∇ετεη|2 | {z } ≤C +|∇εφ|2 −−−→η→0 0, ∀φ ∈ H01(Ω).
Moreover, vε and τε satisfy the following estimate:
rνρd dt |v ε 1|2+ |εvε2|2 +λ ∗ 2ε d dt |τ ε 11|2+ 2|τ12ε |2+ |τ22ε|2 +1 2r(1 − r)ν 2 |∂ xvε1|2+ 1 ε∂zv ε 1 2 + |ε∂xvε2|2+ |∂zv2ε|2 ! +1 2 1 ετ ε 11 2 + 1 ετ ε 12 2 +1 2 1 ετ ε 22 2 ≤ C. (5.18) It remains to pass to the limit as ε tends to zero. After integrating (5.18) between 0 and T , it yields that
⊲ kvε1kL2(L2)≤ k∂zvε1kL2(L2)≤ Cε, thus the following convergences hold in L2(0, T, L2(Ω)) as
ε tends to zero:
From these convergences, it follows that uε
1 = u∗1+ v1ε→ u∗1 in L2(0, T, L2(Ω)) and ∂zuε1 → ∂zu∗1
in L2(0, T, L2(Ω)).
⊲ k∂xvε1kL2(L2) ≤ C, thus ∂xv1ε converges weakly in L2(0, T, L2(Ω)). Now, since it is already
known that uε
1→ u∗1, it follows that ∂xuε1 ⇀ ∂xu∗1 in L2(0, T, L2(Ω)).
⊲ Similarly kv2εkL2(L2) ≤ k∂zvε2kL2(L2) ≤ C, thus εv2ε and ε∂zvε2 converge strongly to zero in
L2(0, T, L2(Ω)), and thus uε2 = εu∗
2+ εvε2→ 0 in L2(0, T, L2(Ω)), and ∂zuε2 → 0 in L2(0, T, L2(Ω)).
⊲ k∂xvε2kL2(L2)≤ C
ε, thus ∂xu
ε
2 converges weakly in L2(0, T, L2(Ω)). Since uε2 → 0, it implies
that ∂xuε2 ⇀ 0 in L2(0, T, L2(Ω)).
⊲ kτ11ε kL2(L2), kτ12εkL2(L2), kτ22ε kL2(L2) ≤ Cε, therefore τ11ε, τ12ε, τ22ε → 0 in L2(0, T, L2(Ω)).
Thus εσε
11= σ11∗ +τ11ε → σ11∗ in L2(0, T, L2(Ω)), and in the same way εσ12ε → σ12∗ in L2(0, T, L2(Ω)),
εσε
22→ σ22∗ in L2(0, T, L2(Ω)).
⊲ From the terms with the derivatives in time, using the fact that vε|t=0= uε0− u∗∈ L2(Ω)
and τε|
t=0= σ0ε− σ∗ ∈ L2(Ω) are bounded independently of ε, we can conclude that
kvεkL∞
(L2)≤ C and kτεkL∞
(L2) ≤ C√ε.
These estimates and the uniqueness of the limit imply that v1ε and εv2ε converge weakly-* in L∞(0, T, L2(Ω)) toward zero, and that τε converges strongly in L∞(0, T, L2(Ω)) toward zero,
which proves the last estimate (5.16).
Note that in a simplified case (with a simpler geometry), the hypothesis (5.12) is satisfied under a small data assumption on the physical parameters.
Remark 5.5. When h is constant with respect to x, p∗ is also independent of x, so that equation
(4.1) reduces to −(1 − r)∂z2u ∗ 1− r ∂ ∂z ∂zu∗1 1 + λ∗2|∂ zu∗1|2 = 0.
It has been shown in [8] for example that for r < 8/9 this equation admits a unique solution u∗
1 = s(1 −hz).
Now, it follows that σ∗ 12 = rν∂zu∗1 1 + λ∗2|∂ zu∗1|2 = −rνs h + λ∗2s2/h, and σ ∗ 11 = −σ∗22 = −λ∗∂zu∗1σ12∗ = −rνs2λ∗ h2+ λ∗2s2.
In this case, hypothesis (5.12) becomes more simple. Since ∂zu∗1= −s/h, σ11∗ and σ12∗ are constant
with respect to z, so that the last two conditions are trivially verified. Using the fact that r < 8/9, it leads to a smallness condition on sλ∗ with respect to h (sλ∗
≤ h/12 is enough in order to satisfy all conditions).
5.3 Convergence of the pressure
It remains to prove the convergence of the pressure.
Theorem 5.6. Under the same smallness assumption (5.12), the following convergence result holds up to a subsequence for p:
ε2p →
ε→0p ∗
in D′(0, T, L2(Ω)). (5.20)
Proof. Throughout the proof, C will denote some generic constants independent of ε. Let ε ≤ 1. Let us integrate over ΩT = Ω × (0, T ) equation (5.1a) multiplied by ε2ϕ1, for any function
φ1∈ H01(Ω). It follows: ρε2 Z ΩT ∂tv1φ1+ ρε2 Z ΩT v1∂xv1φ1+ ρε Z ΩT v2∂zv1φ1+ (1 − r)νε2 Z ΩT ∂xv1∂xφ1+ (1 − r)ν Z ΩT ∂zv1∂zφ1 + Z ΩT ∂xq φ1= −ε Z ΩT τ11∂xφ1− Z ΩT τ12∂zφ1+ ε2 Z ΩT L1φ1+ ε Z ΩT C1φ1, ∀φ1∈ H01(Ω). (5.21) Using the fact that φ1 is independent of t, the first term becomes
ρε2 Z ΩT ∂tv1φ1= ρε2 Z Ω φ1 T Z 0 ∂tv1 = ρε2 Z Ω φ1(v1(T ) − v1(0)),
where v1(0) = u10− u∗1 denotes the value of v1 at time t = 0. Now, introducing
π =
T
Z
0
q dt,
and using integration by parts for the pressure term (the boundary term is zero since φ1 ∈ H01(Ω)),
(5.21) becomes: ∀φ1∈ H01(Ω), ρε2 Z Ω φ1(v1(T ) − v1(0)) + ρε2 Z ΩT v1∂xv1φ1+ ρε Z ΩT r2∂zv1φ1+ (1 − r)νε2 Z ΩT ∂xv1∂xφ1 + (1 − r)ν Z ΩT ∂zv1∂zφ1− Z Ω π ∂xφ1= −ε Z ΩT τ11∂xφ1− Z ΩT τ12∂zφ1+ ε2 Z ΩT L1φ1+ ε Z ΩT C1φ1.
leads: Z ΩT v1∂xv1φ1 ≤ |φ1|δ′ T Z 0 |v1|2+δ|∂xv1|, (5.22) where 1 2 + δ+ 1 2 + 1
δ′ = 1 (which implies that δ
′ = 2(2 + δ)
δ ). According to interpolation theory,
L2, L4θ = L2+δ for θ = δ
2 + δ, and the following estimate holds: |v1|2+δ ≤ C|v1|θ4|v1|1−θ.
Moreover Lemma 3.2 of [1] states that for v1 ∈ H01(Ω), it holds:
|v1|4≤
√
2|∂xv1|1/4|∂zv1|3/4.
Using the two last inequalities and Poincar inequality, (5.22) becomes
ρε2 Z ΩT v1∂xv1φ1 ≤ ρε2|φ1|δ′C T Z 0 |∂xv1|θ/4|∂zv1|3θ/4|∂zv1|1−θ|∂xv1|,
and Hlder inequality implies that ρε2 Z ΩT v1∂xv1φ1 ≤ ρε2|φ1|δ′Ck∂xv1k1+θ/4 L2(Ω T)k∂zv1k 1−θ/4 L2(Ω T).
Now, choose θ (and thus δ) such that δ′
≥ 6. It suffices to take θ ≤ 13, for example take θ =
1 3. Then δ′ = 6, and the usual Sobolev embeddings read H1(Ω) ֒→ L6(Ω) (which is true in dimension
2 or 3). Therefore, the last estimate becomes ρε2 Z ΩT v1∂xv1φ1 ≤ ρε2Ckφ1kH1k∂xv1k13/12 L2(Ω T)k∂zv1k 11/12 L2(Ω T).
Now, recalling that k∂zv1kL2(L2)≤ Cε and k∂xv1kL2(L2)≤ C, we conclude
ρε2 Z
ΩT
v1∂xv1φ1≤ ρε2Ckφ1kH1ε11/12= ρε2+11/12Ckφ1kH1 ≤ Cεkφ1kH1.
In a similar way, it holds ρε
Z
ΩT
r2∂zv1φ1 ≤ ρε2−1/12Ckφ1kH1 ≤ eCεkφ1kH1.
For the term ρε2R
Ω
for v1(T ), we use Poincar inequality. It follows, using the fact that |∂zv1| ≤ Cε:
ρε2 Z
Ω
φ1(v1(T )−v1(0)) ≤ (C|v1|+C)ε2kφ1kH1 ≤ (C|∂zv1|+C)ε2kφ1kH1 ≤ Cε2kφ1kH1 ≤ Cεkφ1kH1.
For the other linear terms, a simple application of Cauchy-Schwarz inequality allows to obtain similar estimates. Indeed, it suffices to use the estimate (5.18) in order to estimate the L2-norm
of ∂xv1, ∂zv1, τ11, τ12, L1, C1. For example, since |∂xv1| ≤ C, the following estimate holds:
ρε2 Z
Ω
∂xv1∂xφ1 ≤ ρε2|∂xv1| |∂xφ1| ≤ Cε2kφ1kH1.
For the terms L1 and C1, C1 and the constant part of L1 are obviously bounded uniformly in
ε. It remains to estimate the linear term L1 of L1. Recalling its definition and using Poincar
inequality in the second estimate:
|L1| ≤ C (|v1| + |v2| + |∂xv1| + |∂zv1|) ≤ C (|∂zv1| + |∂xv1| + |∂zv2|) .
Using again (5.18), the boundedness of L1 follows:
|L1| ≤ C. Hence ∀φ1∈ H01(Ω): Z Ω ∂xπ φ1 ≤ C ε + ε2|∂xv1| + |∂zv1| + ε|τ11| + |τ12| + ε2|L1| + ε|C1| kφ1kH1 ≤ Cεkφ1kH1.
The same approach with (5.1b) gives a similar estimate, for all φ2∈ H01(Ω):
Z
Ω
∂zπ ϕ2 ≤ C ε + ε4|∂xv2| + ε2|∂zv2| + ε2|τ12| + ε|τ22| + ε2|L2| + ε|C2|kφ2kH1 ≤ Cεkφ2kH1.
Thus we can conclude that k∇πkL∞
(H−1)≤ Cε.
Now recall that for f ∈ L20(Ω), it holds that |f| ≤ k∇qkH−1 (see for example [17]). Since
p ∈ L2
0(Ω) and p∗ ∈ L20(Ω), q lies in L20(Ω). From the definition of π as function of q, it is clear
that π ∈ L20(Ω).
This allows to deduce
|π|L∞(L2)≤ k∇πkL∞(H−1)≤ Cε → 0,
thus π tends to zero in L∞(0, T, L2
0(Ω)) when ε → 0. Now, since q =
∂π
This finishes the proof.
5.4 Open problems
This work concerns only the solutions of the problem (3.1) that are obtained as the limit of the regularized problem we chose (with an additional term −η∆σ). Since there is no uniqueness result for problem (3.1), it is not known how other solutions behave.
Formally, the passing to the limit can be done for a 6= 0 (see [4]), and a similar limit problem (involving the parameter a, but of the same structure). However, the proof of the existence theorem in ˆΩε strongly relies on the fact that a = 0. No global results are proved in the case a 6= 0.
Last, since the computations are independent of the dimension of the domain Ω, the result should be true in the three-dimensional case. The limit problem on (u∗, p∗, σ∗) reads:
(1 − r)ν∂z2u ∗ 1− ∂xp∗+ ∂zσ13∗ = 0, (1 − r)ν∂z2u∗2− ∂xp∗+ ∂zσ23∗ = 0, ∂zp∗ = 0, ∇ · u∗ = 0, −λ∗∂zu∗1σ13∗ + σ11∗ = 0, −λ ∗ 2 ∂zu ∗ 1σ∗13− ∂zu∗2σ23∗ + σ12∗ = 0, −λ∗∂zu∗2σ ∗ 23+ σ ∗ 22= 0, λ∗ 2 ∂zu ∗ 2(σ ∗ 33− σ ∗ 22) − λ∗ 2 ∂zu ∗ 1σ ∗ 12+ σ ∗ 23= rν∂zu∗2, λ∗(∂zu∗1σ∗13+ ∂zu∗2σ∗23) + σ33∗ = 0, λ∗ 2 ∂zu ∗ 1(σ∗33− σ11∗ ) − λ∗ 2 ∂zu ∗ 2σ12∗ + σ12∗ = rν∂zu∗1. (5.23)
Acknowledgements
The authors would like to thank P. Mironescu for useful discussions concerning section 4.3 of this paper.
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